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Stability boundary in the plane (μ, p) for case (px) 1 in Table 3, when a u (x) is constant (solid), linear (dashed) and quadratic (dotted). Different rows correspond to different types of v-dependence, see also Table 2. Stability region is below the boundary
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We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on ce...
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Citations
... The literature on functional differential equations with delay presents some works on integro-differential and integral problems with time integral intervals depending of the state. In particular, we mention the interesting papers [4,10,11,33,34], where some finite-dimensional population models with SDD are studied via an integral equation with state-dependent integration intervals. We also note the papers [6,24,32,36] on integral and integro-differential abstract problems in finite-dimensional spaces and the interesting papers by Angelov et al. [1][2][3], where some finite-dimensional neutral explicit differential equations with SDD are studied via integral equations with state-dependent integration intervals. ...
In this work, we study a new class of integro‐differential equations with delay, where the informations from the past are represented as an average of the state over state‐dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro‐differential equations arising in the theory of population dynamics.
... As a consequence, local bifurcations for SDDEs, related to singularities of codimension-two or higher, are still hard to be analytically located since they require C k -smooth local center manifolds with k > 1. More considerable efforts have been done recently to overcome many difficulties in the theory of SDDEs [Krisztin, 2003;Walther, 2004;Hartung et al., 2006;Qesmi & Walther, 2009;Hu & Wu, 2010;Mallet-Paret & Nussbaum, 2011;Stumpf, 2012;Sieber, 2012;Getto et al., 2019;Calleja et al., 2017;Eichmann, 2006]. Moreover, the established theoretical results have been used with some success in many fields of application. ...
Dynamic behavior investigations of infectious disease models are central to improve our understanding of emerging characteristics of model states interaction. Here, we consider a Susceptible-Infected (SI) model with a general state-dependent delay, which covers an immuno-epidemiological model of pathogen transmission, developed in our early study, using a threshold delay to examine the effects of multiple exposures to a pathogen. The analysis in the previous work showed the appearance of forward as well as backward bifurcations of endemic equilibria when the basic reproductive ratio R0 is less than unity. The analysis, in the present work, of the endemically infected equilibrium behavior, through the study of a second order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation on the upper branch of the backward bifurcation diagram and gives the criteria for stability switches. Furthermore, the inclusion of state-dependent delays is shown to entirely change the dynamics of the SI model and give rise to rich behaviors including periodic, torus and chaotic dynamics.
... Let be the generator of the equation (4). Via (3) we have (7) where via (4) Using Lemma for with , via (5) we have (8) Similarly, using Lemma for with , via (5) we obtain (9) From it follows that (10) Choosing the additional functional in the form we obtain (11) From (10), (11) for the functional we have ...
... Let be the generator of the equation (4). Via (3) we have (7) where via (4) Using Lemma for with , via (5) we have (8) Similarly, using Lemma for with , via (5) we obtain (9) From it follows that (10) Choosing the additional functional in the form we obtain (11) From (10), (11) for the functional we have ...
This chapter is devoted to stability investigation of systems with state-dependent delays under stochastic perturbations. Sufficient conditions of asymptotic mean square stability for the zero solution of a linear stochastic differential equation with distributed delays are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Besides delay-independent and delay-dependent conditions of stability in probability are obtained for two equilibria of a nonlinear stochastic differential equation with delay and exponential nonlinearity. The negative definiteness of matrices in the obtained LMIs is checked using the special MATLAB program. It is noted that the proposed research method can be used for the study of other types of linear and nonlinear systems with state-dependent delays. Numerical simulation of solutions of the considered stochastic differential equations with state-dependent delays illustrate the presented here theoretical results and open to readers attention a new unsolved problem of the obtained stability conditions improving.
... The cell SD-DDE (1.1-1.4) may feature a unique positive equilibrium emerging from the zero equilibrium in a transcritical bifurcation: q may decrease to a negative value and q(0) should increase from negative to positive upon variation of the bifurcation parameter, see [6,8]. A combination of the discussed results of [13,19,7] facilitated a local stability analysis of equilibria for the cell SD-DDE in [8]. ...
... The cell SD-DDE (1.1-1.4) may feature a unique positive equilibrium emerging from the zero equilibrium in a transcritical bifurcation: q may decrease to a negative value and q(0) should increase from negative to positive upon variation of the bifurcation parameter, see [6,8]. A combination of the discussed results of [13,19,7] facilitated a local stability analysis of equilibria for the cell SD-DDE in [8]. ...
... The paper [8] contains numerical and analytical evidence that the interior equilibrium is stable upon emergence and destabilizes in a Hopf bifurcation. The latter motivates also analytical research for periodic solutions for the cell SD-DDE. ...
We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.
... With an eye on future work, it is reasonable to expect that the results of the present work can be combined with those of [22] for RFDE to obtain an analogous theory for systems of coupled RE and RFDE, similarly to what is done in [21, section 4] for equilibria (although some difficulties may arise from the coupling). This would represent a further step towards the dynamical analysis of complex yet realistic models, e.g., those recently proposed for modeling physiologically structured populations [21,24,29,50]. Establishing this theory for coupled RE and RFDE would also benefit the numerical method already developed in [39], which combines and extends the ideas of [5] for RFDE and of [4] for RE. ...
... The initial value problem defined by (29) for σ > τ and (z k ) τ = ψ k is equivalent to 5 ...
We prove the validity of a Floquet theory and the existence of Poincaré maps for periodic solutions of renewal equations, also known as Volterra functional equations. Our approach is based on sun–star perturbation theory of dual semigroups and relies on a spectral isolation property and on the regularity of the semiflow. This contributes a new chapter to the stability analysis, in analogy with ordinary and retarded functional differential equations as well as the case of equilibria.
... From a modelling perspective the cell cycle duration is a positive time delay between two sequential cell proliferation events. There are two main types of models that incorporate time delays: one involves functional differential equations (Mackey and Rudnicki 1994;Byrneo 1997;Baker et al. 1997Baker et al. , 1998Villasana and Radunskaya 2003;Getto and Waurick 2016;Getto et al. 2019;Cassidy and Humphries 2020), of which delay differential equations are a specific type; and multi-stage models (Yates et al. 2017;Simpson et al. 2018;Vittadello et al. 2018Vittadello et al. , 2019Gavagnin et al. 2019). Models incorporating time delays are consistent with the kinetics of cell proliferation, and can result in a better qualitative and quantitative fit of the model to experimental data (Baker et al. 1998). ...
We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the nonnegativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations.
... In [4] the idea is launched to first reduce the infinite dimensional dynamical system corresponding to a delay equation to a finite dimensional one by pseudospectral approximation, and next use tools for ordinary differential equations (ODE) in order to perform a numerical bifurcation analysis. Several examples illustrate that this approach is promising (also see [5,6,7,8,9,10,11]). ...
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.
... In the following we take a famous example from mathematical biology, namely the 'Nicholson's blowflies' equation, as a testing ground to illustrate some features of the approach. However, we remark that the methodology presented here (pseudospectral approximation combined with software for bifurcation analysis of ODE) can be applied in a much more general setting: it is indeed a promising procedure to study differential equations with distributed, state-dependent, and even infinite delays [6,22,19], as well as nonlinear renewal equations [7] and first order partial differential equations [31]. The advantage of considering Nicholson's blowflies equation in this context is due to the fact that explicit comparisons are possible, both with analytically computed quantities and with alternative numerical approximations, as will become clear later on. ...
... The real part of the denominator of (A.21) is non-zero for ω = 0; hence the denominator of (A.21) is non-zero for ω = 0, which means that ω = 0 is a simple zero of (A.17) for b 1 , b 2 defined in (A. 19). The real part of (A.21) becomes Re λ (ω) = 14 − 2b 1 (ω) 4ω 2 + (6 − 2b 1 (ω)) 2 . ...
... On the interval (−4, 4) the expression for b 1 in (A.19) attains its maximum b 1 = 1 for ω = 0. Therefore Re λ (ω) = 0 along the curve (A. 19)-(A.20). Moreover, since A 2 has exactly three eigenvalues (counting multiplicity), the non-resonance condition is in this case easy to check. ...
... Time delay has been included in numerous models in biology, with applications in biochemical negative feedback [15], cell growth and division [1,11], or cell maturation [10], but are less common in biomechanics. In our case we introduce this delay in an evolution law of the cell or tissue rest-length. ...
Cells and tissues exhibit oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with cell biochemical signalling, it is yet unclear the role of these oscillations in triggering drastic cell reorganisation events or instabilities, and the coupling of this oscillatory response with tested visco-elastic properties. We here present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability properties as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilise the oscillatory response. Furthermore, we numerically verify that the extension of the model to non-linear strain measures is able to generate sustained oscillations that alternate between stable and unstable regions.
... The main focus of population dynamics has been a characterization of alterations in the numbers, sizes and age distribution of individuals and of potential internal or external causes provoking these changes. In the studies of structured population equations, linear semigroup methods were successfully developed to investigate the linear stability regularity and bifurcation phenomena of solutions for linearized systems, see [16,26,31,37]. The last years have witnessed an invigorated interest in age/size-structured population dynamics due to the wide applications. ...
This work focuses on the long time behavior for a size-dependent population system with diffusion and Riker type birth function. Some dynamical properties of the considered system is investigated by using \begin{document}\end{document}-semigroup theory and spectral analysis arguments. Some sufficient conditions are obtained respectively for asymptotical stability, asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system. In the end several examples and their simulations are also provided to illustrate the achieved results.
